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Theorem lecmtN 32792
Description: Ordered elements commute. (lecmi 27254 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
lecmt.b  |-  B  =  ( Base `  K
)
lecmt.l  |-  .<_  =  ( le `  K )
lecmt.c  |-  C  =  ( cm `  K
)
Assertion
Ref Expression
lecmtN  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  X C Y ) )

Proof of Theorem lecmtN
StepHypRef Expression
1 omllat 32778 . . . . 5  |-  ( K  e.  OML  ->  K  e.  Lat )
213ad2ant1 1026 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
3 simp2 1006 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
4 omlop 32777 . . . . . . 7  |-  ( K  e.  OML  ->  K  e.  OP )
543ad2ant1 1026 . . . . . 6  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  OP )
6 lecmt.b . . . . . . 7  |-  B  =  ( Base `  K
)
7 eqid 2422 . . . . . . 7  |-  ( oc
`  K )  =  ( oc `  K
)
86, 7opoccl 32730 . . . . . 6  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
95, 3, 8syl2anc 665 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( oc `  K ) `  X
)  e.  B )
10 simp3 1007 . . . . 5  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
11 eqid 2422 . . . . . 6  |-  ( join `  K )  =  (
join `  K )
126, 11latjcl 16297 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )
132, 9, 10, 12syl3anc 1264 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )
14 lecmt.l . . . . 5  |-  .<_  =  ( le `  K )
15 eqid 2422 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
166, 14, 15latmle1 16322 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )  -> 
( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  X )
172, 3, 13, 16syl3anc 1264 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  X )
186, 15latmcl 16298 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) Y )  e.  B )  -> 
( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  e.  B )
192, 3, 13, 18syl3anc 1264 . . . 4  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  e.  B )
206, 14lattr 16302 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( X (
meet `  K )
( ( ( oc
`  K ) `  X ) ( join `  K ) Y ) )  e.  B  /\  X  e.  B  /\  Y  e.  B )
)  ->  ( (
( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  X  /\  X  .<_  Y )  ->  ( X (
meet `  K )
( ( ( oc
`  K ) `  X ) ( join `  K ) Y ) )  .<_  Y )
)
212, 19, 3, 10, 20syl13anc 1266 . . 3  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( X ( meet `  K
) ( ( ( oc `  K ) `
 X ) (
join `  K ) Y ) )  .<_  X  /\  X  .<_  Y )  ->  ( X (
meet `  K )
( ( ( oc
`  K ) `  X ) ( join `  K ) Y ) )  .<_  Y )
)
2217, 21mpand 679 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  -> 
( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  Y ) )
23 lecmt.c . . 3  |-  C  =  ( cm `  K
)
246, 14, 11, 15, 7, 23cmtbr4N 32791 . 2  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X C Y  <-> 
( X ( meet `  K ) ( ( ( oc `  K
) `  X )
( join `  K ) Y ) )  .<_  Y ) )
2522, 24sylibrd 237 1  |-  ( ( K  e.  OML  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  ->  X C Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   class class class wbr 4423   ` cfv 5601  (class class class)co 6306   Basecbs 15121   lecple 15197   occoc 15198   joincjn 16189   meetcmee 16190   Latclat 16291   OPcops 32708   cmccmtN 32709   OMLcoml 32711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-preset 16173  df-poset 16191  df-lub 16220  df-glb 16221  df-join 16222  df-meet 16223  df-lat 16292  df-oposet 32712  df-cmtN 32713  df-ol 32714  df-oml 32715
This theorem is referenced by:  cmtidN  32793  omlmod1i2N  32796
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